Add butter and sugar to a stand mixer. Cream the ingredients on high speed until well blended. Reduce the speed to medium low and add the egg yolk. Mix in the egg until fully incorporated and then add the vanilla.
Once the wet ingredients are mixed in, add the flour and salt to the mixture. After the dough is completely mixed, cover the bowl with plastic wrap and place in the refrigerator for at least 2 1/2 hours to allow the dough to set.
Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper or spray with cooking spray. Using your hands, roll the cookie dough into balls (the recipe should make around 1 dozen cookies if you use about a tablespoon of dough for each cookie).
Place the cookies in the oven for 3 minutes and then remove from the oven. The dough will now be soft enough to press with your thumb. Using your thumb, make an indentation on the top of each cookie. Fill each cookie with 1/2 teaspoon of jelly, or the amount desired.
Bake the cookies for another 12-15 minutes or until lightly browned on the edges. Allow cookies to cool for 30 minutes before icing.
While the cookies are cooling, you can make the icing. In a small bowl, stir the powdered sugar, half and half and vanilla. If the icing is too thick, add a little more half and half, if it is too runny add more powdered sugar until you reach a desired consistency. Once the cookies have cooled, drizzling the icing over them.